Similar Triangles and Slope CC8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non‐vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Similar Triangles • Similar Triangles are triangles who have the same shape, but not necessarily the same size. The corresponding angles of similar triangles are congruent and their corresponding sides are in PROPORTION. The similar triangles increase or decrease at a constant rate. How do I know if two triangles are similar? . If two triangles are similar, the cross products of their corresponding sides are equal. 5 3 10 6 10 3 5 6 5 units 30 30 Since the cross products of the corresponding sides are equal, the triangles are similar. 10 units 3 units 6 units Rates of Proportionality in a Triangle? Make a rate of the legs in each of these right triangles and compare the results. When making your rate, compare the vertical leg (rise) to the horizontal leg (run). What did you notice? 4 The red triangle has a rate of 4 to 8 or 8 4 5 The blue triangle has a rate of 5 to 10 or 10 8 5 10 3 The green triangle has a rate of 3 to 6 or 6 3 6 How many triangles do you see? Find the ratio of vertical to horizontal leg of each triangle. Then simplify to a fraction. The simplified fraction should be the SLOPE of the red line. 3 3 7 7 9 9 The SLOPE of the red line is 1 because all of the slope ratios simplify to 1. Coordinate Plane/Ordered Pairs The rate of each triangle can be simplified to ½ ! What do you notice about these triangles and their hypotenuse in the illustration below? 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑙𝑒𝑔 𝑅𝑖𝑠𝑒 𝑜𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑙𝑒𝑔 𝑅𝑢𝑛 Positive slope Negative slope Rises from left to right Falls from left to right 𝑜𝑟 3 2 rise 2 run 3 Zero slope Undefined slope Horizontal line Vertical line rise 0 0 run 5 rise 5 Undefined run 0 Draw triangles to find the slope of the line. The slope of the red line is negative since the triangles are moving down. For the smaller triangle, the vertical change is 2 and the horizontal change is 3. For the larger triangle, the vertical change is 4 and the horizontal change is 6. 4 The slope for the red line must be 6 or 2 . 3

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